Grey atmosphere is a useful set of approximations made for radiative transfer applications. The application of the grey atmosphere approximation is the primary method astronomers use to determine the temperature and basic radiative properties of astronomical objects including the sun, planets with atmospheres, other stars, and interstellar clouds of gas and dust. The primary approximation is assumption that the absorption coefficient, typically represented by an α, has no wavelength or frequency dependence for the frequency range being worked in. Typically a number of other assumptions are made simultaneously: (1) the atmosphere has a plane-parallel atmosphere geometry and (2) the atmosphere is in a thermal radiative equilibrium. This set of assumptions leads directly to the mean intensity

Radiance

Radiance and spectral radiance are radiometric measures that describe the amount of radiation such as light or radiant heat that passes through or is emitted from a particular area, and falls within a given solid angle in a specified direction. They are used to characterize both emission from...

 and source function

Source function

The source function is a characteristic of a stellar atmosphere, and in the case of no scattering of photons, describes the ratio of the emission coefficient to the absorption coefficient. It is a measure of how photons in a light beam are removed and replaced by new photons by the material it...

 being directly equivalent to a blackbody Planck function of the temperature at that optical depth

Optical depth

Optical depth, or optical thickness, is a measure of transparency. Optical depth is defined by the negative logarithm of the fraction of radiation that is not scattered or absorbed on a path...

.

Integrating the first and second moments of the radiative transfer equation, applying the above relation and the Two-Stream Limit approximation leads to information about each of the higher moments. The first moment of the mean intensity is constant regardless of optical depth

Optical depth

Optical depth, or optical thickness, is a measure of transparency. Optical depth is defined by the negative logarithm of the fraction of radiation that is not scattered or absorbed on a path...

:



The second moment of the mean intensity is then given by:



Note that the Eddington approximation is a direct consequence of these assumptions.

Defining an effective temperature for the Eddington flux and applying the Stefan-Boltzmann law, realized this relation between the externally observed effective temperature and the internal blackbody temperature of the medium.



The results of the grey atmosphere solution: The observed temperature is a good measure of the true temperature at an optical depth and the surface temperature is . This approximation makes the source function linear in optical depth.

External links

• www.astro.uu.se/~ke/summ9.pdf
• http://xweb.geos.ed.ac.uk/~stephan/pdf/lect_PhysClim2003_07_Notes.pdf